Definition and characterization

Definition 10.1.1   Let $ f$ be a mapping defined over a domain $ D$ in the complex plane. and let be given $ z_0 \in D$ .

If $ f$ preserves the size and the sense of the angle of intersection of any two curves intersecting at $ z_0$ , then $ f$ is called conformal at $ z_0$ .

If $ f$ is conformal at any point of the domain $ D$ , it is called conformal in $ D$ .

Figure 1: Conformal mapping.
\begin{figure}\centering
\mbox{
\subfigure[]{\epsfig{file=Conformal-1.eps,height...
...\qquad
\subfigure[]{\epsfig{file=Conformal-2.eps,height=4.5cm}}
}\end{figure}

Theorem 10.1.2   Let $ f$ be a function analytic over a domain $ D$ . If $ z_0$ is a point of $ d$ where $ f'(_0) \neq 0$ , the $ f$ is conformal at $ z_0$ .



Noah Dana-Picard 2007-12-24